# Solving Exponential and Logarithmic Equations

**I. Review of Some Key Properties of Exponential and
Logarithmic Functions**

A. Base-Exponent Property

if and only if x = y

B. Inverse Properties

Exponential functions and logarithmic functions are inverses which undo one
another.

C. Change of Base Formula

D. Property of Logarithmic Equality

if and only if x = y.

E. Product Rule of Logarithms

F. Quotient Rule of Logarithms

G. Power Rule of Logarithms

**II. Exponential Equations**

A. Definition

Equations with variables in the exponents are called exponential equations.

Examples:

B. Strategies for Solving Exponential Equations

Method 1: Get the Same Base

Rewrite each side of the equation as a power of the same base and then apply the

Base-Exponent Property to solve for the variable. To check, plug your un-rounded

solution into the original equation and see if it works.

**Example 1 **

Check:

Method 2: Using to
undo

Isolate the exponential expression. Take the log base a of both sides. Use the
inverse

property to simplify one side and, if necessary, the change of base formula to
simplify

the other. Solve for the variable. To check, plug your un-rounded solution into
the

original equation and see if it works.

**Example 2**

Method 3: Using ln and the Power Rule to undo

Isolate the exponential expression. Take the natural log of both sides. Use the
power

rule of logarithms to make the exponent a coefficient. Solve for the variable.
To check,

plug your un-rounded solution into the original equation and see if it works.

**Example 3**

Method 4: Graphical Solution

Put the equation in standard form (everything on the left side of the equal sign
and zero

on the right). Let the left side equal .
Adjust the window. Use the Zero / Root

feature to find the zeros.

**Example 4**

**III. Logarithmic Equations**

A. Definition

Equations containing variables in logarithmic expressions are called logarithmic
equations.

Examples:

B. Strategies for Solving Logarithmic Equations

Method 1: Using the Definition of a Logarithm

Use the properties of logarithms to write the equation so
there is no more than one log

on each side. Use the definition of a log to rewrite the equation in exponential
form.

Solve for the variable. To check, plug your un-rounded solution into the
original

equation and see if it works. You may have to use the change of base formula.

**Example 5**

Method 2: Exponentiating Both Sides

Use the properties of logarithms to write the equation so there is no more than
one log on

each side. Exponentiate both sides using base a. Simplify one side using the
inverse

property to solve for the variable. To check, plug your un-rounded solution into
the original

equation and see if it works. You may have to use the change of base formula.

**NOTE:** It is especially important to check your solutions if you end up
with a quadratic.

**Example 6**

Method 3: Graphical Solution

Put the equation in standard form (everything on the left side of the equal sign
and zero

on the right). Let the left side equal y_{1}. If the logs are not base 10 or base
e, use the

change of base formula to rewrite the logs. Adjust the window. Use the Zero /
Root

feature to find the zeros.

**Example 7 **2ln(x–5.1) =8.9